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Higher order hidden Markov models for time series
(2021) Skapera, Antonia; Σκαπέρα, Αντωνία; Athens University of Economics and Business, Department of Statistics; Livada, Alexandra; Vrontos, Ioannis; Besbeas, Panagiotis
A hidden Markov model (HMM) is a statistical model in which the system being modelled is assumed to be a Markov process with unobservable (hidden) states. HMMs have found application in a wide variety of disciplines, ranging from signal processing and engineering to finance and the environment. Typically the underlying Markov process in a HMM is assumed to be first-order. We consider the use of higher-order HMMS accommodating longer-range Markov dependence. We focus on second-order dependence, and employ an approach that transforms a second-order HMM into an equivalent first-order. The approach is general to any order, and opens the way to estimating higher order HMMs using standard techniques for first-order models. Despite the theoretical appeal of higher-order HMMs, their larger number of parameters can be detrimental to their performance in practice. We explore this issue, illustrating their practical utility using real world applications. In the first application we consider a binary time-series based on the Old Faithful geyser data. We fit first-and second-order HMMs and choose between them using information criteria. In the second application we consider a famous count time series reflecting the annual number of major earthquakes that happened globally between1900 and 2006. We fit first-and second-order HMMs when the state distribution, the number of latent states, and the nature of the serial dependence, including the true order, are unknown. Because of the highly increasing number of parameters, we explore parameter reduction through fitting mixture transition distribution (MTD) models, which allow second-order dependence, but use less parameters. We compare different models and specifications using information criteria to choose which models fits better the data. Finally, we perform a simulation study using the earthquakes data set, comparing first- and second-order HMMs and MTD- second order HMMs. We conclude that there are differences between the estimated parameters of these models.